Page:A History of Mathematics (1893).djvu/88

 proportion $$\scriptstyle{{a:{{a+b} \over 2}}={{2ab} \over {a+b}}:b}$$. Iamblichus says that Pythagoras introduced it from Babylon.

In connection with arithmetic, Pythagoras made extensive investigations into geometry. He believed that an arithmetical fact had its analogue in geometry, and vice versa. In connection with his theorem on the right triangle he devised a rule by which integral numbers could be found, such that the sum of the squares of two of them equalled the square of the third. Thus, take for one side an odd number $$\scriptstyle{(2n+1)}$$; then $$\scriptstyle{{{(2n+1)^2-1} \over 2}=2n^2+2n}$$=the other side, and $$\scriptstyle{(2n^2+2n+1)}$$=hypotenuse. If $$\scriptstyle{2n+1=9}$$, then the other two numbers are 40 and 41. But this rule only applies to cases in which the hypotenuse differs from one of the sides by 1. In the study of the right triangle there doubtless arose questions of puzzling subtlety. Thus, given a number equal to the side of an isosceles right triangle, to find the number which the hypotenuse is equal to. The side may have been taken equal to $$\scriptstyle{1,~2,~{3 \over 2},~{6 \over 5}}$$, or any other number, yet in every instance all efforts to find a number exactly equal to the hypotenuse must have remained fruitless. The problem may have been attacked again and again, until finally "some rare genius, to whom it is granted, during some happy moments, to soar with eagle's flight above the level of human thinking," grasped the happy thought that this problem cannot be solved. In some such manner probably arose the theory of irrational quantities, which is attributed by Eudemus to the Pythagoreans. It was indeed a thought of extraordinary boldness, to assume that straight lines could exist, differing from one another not only in length,—that is, in quantity,—but also in a quality, which, though real, was absolutely invisible.[7] Need we wonder that the Pythagoreans saw in